Dualization invariance and a new complex elliptic genus

Abstract. We define a new elliptic genus ψ on the complex bordism ring. With coefficients in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>[</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> <m:mo>]</m:mo> </m:mrow> </m:math> $\mathbb {Z}[1/2]$ , we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℙ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>E</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>-</m:mo> <m:mi>ℙ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>E</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $\mathbb {P}(E)-\mathbb {P}(E^\ast )$ of projective bundles and their duals onto a polynomial ring on four generators in degrees 2, 4, 6 and 8. As an alternative geometric description of ψ, we prove that it is the universal genus which is multiplicative in projective bundles over Calabi–Yau 3-folds. With the help of the q -expansion of modular forms we will see that for a complex manifold M , the value <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ψ</m:mi> <m:mo>(</m:mo> <m:mi>M</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> $\psi (M)$ is a holomorphic Euler characteristic. We also compare ψ with Krichever–Höhn's complex elliptic genus and see that their only common specializations are Ochanine's elliptic genus and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>χ</m:mi> <m:mi>y</m:mi> </m:msub> </m:math> $\chi _y$ -genus.